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In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups.

Attributes	Values
rdf:type	yago:WikicatSolvableGroups yago:WikicatSubgroupProperties yago:Abstraction100002137 yago:Group100031264 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 ethnic group yago:WikicatFiniteGroups
rdfs:label	Carter subgroup (en)
rdfs:comment	In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups. (en)
dcterms:subject	Finite groups Solvable groups Subgroup properties
Wikipage page ID	5456815 (xsd:integer)
Wikipage revision ID	1099059144 (xsd:integer)
Link from a Wikipage to another Wikipage	Roger Carter (mathematician) Bernd Fischer (mathematician) Mathematics Mathematische Zeitschrift Hall subgroup Alternating group Finite groups Nilpotent group Dihedral group of order 6 Formation (group theory) Solvable groups Subgroup properties Sylow subgroup Group theory Habilitationsschrift Inner automorphism Cartan subalgebra Cartan subgroup Solvable group Finite group Siberian Mathematical Journal Normalizer condition Springer-Verlag Self-normalizing subgroup Nilpotent residual
sameAs	Carter subgroup Carter subgroup Carter subgroup Carter subgroup
txt	yes (en)
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Harv dbt:Harvtxt dbt:Short_description dbt:Harvp dbt:Harvs dbt:Abstract-algebra-stub
author	Vil'yams, N. N. (en)
id	C/c020590 (en)
last	Vdovin (en)
title	Carter subgroup (en)
year	2006 (xsd:integer) 2007 (xsd:integer)
has abstract	In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups. proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the alternating group A5 of order 60 has no Carter subgroups. Vdovin showed that even if a finite group is not solvable then any two Carter subgroups are conjugate. A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups . For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group is also self-normalizing, and a soluble group is generated by any Carter subgroup and its nilpotent residual . viewed the Carter subgroups as analogues of Sylow subgroups and Hall subgroups, and unified their treatment with the theory of formations. In the language of formations, a Sylow p-subgroup is a covering group for the formation of p-groups, a Hall π-subgroup is a covering group for the formation of π-groups, and a Carter subgroup is a covering group for the formation of nilpotent groups . Together with an important generalization, Schunck classes, and an important dualization, Fischer classes, formations formed the major research themes of the late 20th century in the theory of finite soluble groups. A dual notion to Carter subgroups was introduced by Bernd Fischer in. A Fischer subgroup of a group is a nilpotent subgroup containing every other nilpotent subgroup it normalizes. A Fischer subgroup is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Fischer subgroup: again the nonabelian group of order six provides an example as every non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup . (en)
gold:hypernym	Subgroup
prov:wasDerivedFrom	wikipedia-en:Carter_subgroup?oldid=1099059144&ns=0
page length (characters) of wiki page	5444 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Carter_subgroup
is Link from a Wikipage to another Wikipage of	Roger Carter (mathematician) A-group Locally finite group Formation (group theory) History of group theory Imperfect group Carter group Fischer subgroup
is Wikipage redirect of	Carter group Fischer subgroup
is known for of	Roger Carter (mathematician)
is foaf:primaryTopic of	wikipedia-en:Carter_subgroup

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